Ultrasensitive magnetic field detection using a single artificial atom
Efficient detection of magnetic fields is central to many areas of research and has important practical applications ranging from materials science to geomagnetism. High sensitivity detectors are commonly built using direct current-superconducting quantum interference devices (DC-SQUIDs) or atomic systems. Here we use a single artificial atom to implement an ultrahigh sensitivity magnetometer with a size in the micron range. The artificial atom is a superconducting two-level system at low temperatures, operated in a way similar to atomic magnetometry. The high sensitivity results from quantum coherence combined with strong coupling to magnetic field. By employing projective measurements, we obtain a sensitivity of at 10 MHz. We discuss feasible improvements that will increase the sensitivity by over one order of magnitude. The intrinsic sensitivity of this method to AC fields in the 100 kHz - 10 MHz range compares favourably with DC-SQUIDs and atomic magnetometers of equivalent spatial resolution. This result illustrates the potential of artificial quantum systems for sensitive detection and related applications.
Sensitive physical measurements are an essential component of modern science and technology. Developments in this area follow closely scientific discovery and provide in turn tools for practical applications and new research endeavours. Detection of magnetic fields is an area of wide interest, with various applications including medical imaging, geomagnetics, non-destructive materials evaluation, scanning probe microscopy, and electrical measurements clarke_2004_1 (); budker_2007_AtomicMagnReview (). Magnetic field sensors are also enabling tools for fundamental studies of magnetism dang_2010_MagntoHighSens (), spin dynamics degen_2008_ScanningFieldNV (), and mechanical motion arcizet_2011_NVcoupledRes (); kolkowitz_2012_CohSensResonatorSpin ().
Sensitive tools to detect magnetic fields are diverse and include direct-current superconducting quantum interference devices (DC-SQUIDs) clarke_2004_1 (), atomic magnetometers budker_2007_AtomicMagnReview (), Hall probes chang_1992_ScanningHall (), and magnetostrictive sensors forstner_2012_CavityOptoMagnetometer (). DC-SQUIDs have been established for a long time as very sensitive magnetometers clarke_2004_1 (). In recent years, significant advances in atomic control led to the development of atomic magnetometers, which presently compete with DC-SQUIDs for magnetic field detection and have the convenience of operation in a room temperature environment budker_2007_AtomicMagnReview (). Atomic magnetometers employ ensembles of atoms, with each atom evolving quantum coherently in the field to be measured. A detection method similar to atomic magnetometry can be implemented using nitrogen-vacancy defects (NV centers) in diamond crystals taylor_2008_NVDiamondMagn (). Magnetometers based on single NV centers maze_2008_NanoMagnSingleSpin (); balasubramanian_2008_SingleSpinAmbient () have been demonstrated and shown to have interesting prospects as high spatial resolution high sensitivity detectors. In this paper we demonstrate the use of a single artificial atom as an AC magnetometer. The artificial atom, a micron-sized superconducting ring with Josephson junctions, has been studied extensively for quantum computing applications clarke_2008_rev-sup-qb (). Our work establishes this system as an ultrasensitive magnetic field detector.
The principle of our approach follows closely magnetometry based on vapour cells and NV centers budker_2007_AtomicMagnReview (); taylor_2008_NVDiamondMagn (). For a single spin, precession in a field of induction during time leads to an accumulated phase , with the magnetic moment. With a coherent control pulse, the spin is rotated so that the projection along the magnetic field depends on . A single measurement produces a result , corresponding to the two spin states. For repetitions, the average value of the measurement and the variance is . The minimum magnetic field difference, which can be reliably measured, corresponding to a signal to noise ratio of , is . This can be expressed as , with the repetition time of the state preparation, precession, and measurement procedure described above and the total measurement time. The sensitivity can thus be expressed as the quantity , which has units of . The sensitivity increases with the time as long as the evolution is fully quantum coherent; with decoherence taken into account, the optimum is reached when is of the order of the coherence time.
In magnetometers based on this principle, the two spin states of interest are hyperfine or Zeeman levels in alkali budker_2007_AtomicMagnReview () or ground states in NV centers in diamond taylor_2008_NVDiamondMagn (). Detection is done most commonly using an ensemble of atoms/defects manipulated and detected independently. With atoms, sensitivity is further enhanced by a factor , with the volume and the density of the cloud. The dependence of sensitivity implies that there is a tradeoff between sensitivity and spatial resolution. The same tradeoff also appears in magnetic field sensors based on DC SQUIDs clarke_2004_1 ().
Here we use a single artificial atom to implement an ultrasensitive magnetometer for AC magnetic fields. The artificial atom is a persistent current quantum bit (PCQ) mooij_1999_1 (), operated in this experiment at a temperature of . This type of system has been under intense investigation for applications in quantum information processing clarke_2008_rev-sup-qb (). The PCQ is a superconducting quantum ring, with typical size in the micron range, interrupted by three Josephson junctions (see Fig.1c). The two lowest energy eigenstates of the PCQ are characterized by a persistent current , flowing either anticlockwise or clockwise in the qubit ring. In the basis of the persistent current states, the qubit Hamiltonian is given by where is the magnetic flux applied to the qubit ring, is the flux quantum, and is the minimum energy level splitting of the qubit, which occurs at the symmetry point (see Fig.3a). The effective magnetic moment of the qubit is with the energy-level difference between the excited (e) and ground (g) states and the magnetic field applied to the qubit ring. The magnetic moment is given by , with the qubit transition frequency. For our qubit, characterized by , , and , and operated at , reaches the value . This large magnetic moment enables a large sensitivity, despite the coherence time of the PCQ being shorter than in typical atomic systems budker_2007_AtomicMagnReview (); taylor_2008_NVDiamondMagn ().
To coherently control the quantum state of the PCQ, microwave fields at the transition frequency are applied through an on-chip waveguide terminated in a low inductance line (see Fig.1a and b). In a frame rotating at the qubit transition frequency, the microwave field acts as a fictitious magnetic field that induces rotation of the qubit around an axis in the plane; the orientation of the rotation axis in this plane depends on the phase of the driving field. In the same frame, a change in the magnetic field applied perpendicularly to the qubit loop results in a fictitious magnetic field along the axis. Below, we use to denote a rotation of angle around an axis defined by the vector .
Quantum measurement of the PCQ is done by using a circuit-quantum electrodynamics setup wallraff_2004_1 (); abdumalikov_2009_flux-qubit (); niemczyk_2010_cqedstrong (). The qubit is inductively coupled to a superconducting resonator, with a resonance frequency , significantly lower than the qubit transition frequency , and quality factor . A microwave readout pulse of duration and frequency is sent to the qubit. The complex amplitude of the transmitted pulse, as determined in a homodyne measurement wallraff_2005_1 (), is averaged over the duration of the readout pulse. In Fig. 2 we present the results of the qubit measurement. We only show one quadrature of the transmitted voltage, ; the axis for this quadrature is chosen such that it optimizes the measured signal (see Fig. 1a). The qubit is prepared either in the ground state by allowing for a waiting time much longer than the energy relaxation time (Fig 2a) or in the excited state by including a pulse (Fig 2b). The distribution of the values of the homodyne voltage for repetitions of the sequence is shown in Fig. 2c and 2d respectively. The distribution is bimodal, with the two modes corresponding to the qubit energy eigenstates. A threshold can be used to separate the distribution into a part labeled and the complementary part labeled . The threshold is chosen so that it optimizes the readout contrast, which is the difference of the conditional probabilities . The maximum contrast is . This high readout fidelity is essential for the sensitivity of the detector.
In typical magnetometers, as introduced above, free precession in a magnetic field is employed. This procedure is adapted to detection of low-frequency fields, ranging from DC to the inverse of the repetition time, . The sensitivity is proportional to , where is the Ramsey coherence time yoshihara_2006_1 (). This coherence time is short in the PCQ due to the presence of low-frequency magnetic flux noise yoshihara_2006_1 (); bylander_2011_noisePCQ (). Low-frequency noise sets the ultimate limit for measurement of low frequency fields, a situation also encountered for DC-SQUIDs clarke_2004_1 (). For this reason, we focus here on detection of AC magnetic fields. The procedure is illustrated in Fig. 4a. A sequence of pulses (also called a spin-echo sequence) is applied to the qubit at times , , and respectively . The qubit phase precession is given by . The acquired phase is optimized when the frequency of the detected field is equal to . The coherence time during the spin-echo sequence, , is significantly longer than the Ramsey time yoshihara_2006_1 (), which renders the sensitivity to AC field higher than for low frequency fields.
We performed measurements of the evolution of the qubit with a spin-echo sequence of varying total time with an AC magnetic field of frequency applied to the qubit in phase with the spin-echo sequence, as shown in Fig. 4a. The AC magnetic field is applied through the same control line as used for qubit excitation (see Fig. 1a), by applying a voltage of amplitude . The average value of the homodyne voltage is shown in Fig. 3b as a function of the spin-echo sequence time and the voltage amplitude . We observe oscillations as a function of the time , with a frequency , which is proportional to the amplitude (see Fig. 3c).
We next proceed to the characterization of magnetic field detection using the protocol illustrated in Fig. 4a. The output of the detector is the binary signal . The noise in reflects the stochastic nature of quantum measurement; in atomic magnetometry this noise is termed projection noise budker_2007_AtomicMagnReview (). In the following we express the results of detection in terms of the magnetic flux applied to the qubit, as this facilitates the comparison with DC-SQUIDs. The noise in results in an equivalent noise in characterized by the spectral density . Here is the single-sided spectral density of the noise in the qubit readout and is the transfer function of the detector, with the average value of . The transfer function can be expressed as where is determined from the spin-echo measurements shown in Fig. 3c and is determined from qubit spectroscopy (see Fig. 3a). The conversion factor is determined from the measurements shown in Fig. 4b. In this way we determine the equivalent detector input noise fully from a set of measurable quantities, without any assumption on coupling of the field to the qubit. We use a spin-echo control sequence time chosen to correspond approximately to the optimal value for the measured qubit coherence time. We note that in this experiment sample coherence was affected by a two-level fluctuator, as shown by the structure of the spectroscopy peak and also by the fact that there are discrepancies between the observed spin-echo decay and the expected Gaussian law yoshihara_2006_1 (). We use nevertheless the dependence to fit the envelope of the spin-echo oscillations and extract . Figure 4c shows , as obtained by taking the power spectral density of the vs time signal, and the calculated equivalent input detector noise . A theoretical calculation of the sensitivity taking into account the finite measurement fidelity and the experimentally characterized decoherence during the spin echo sequence predicts a flat noise spectrum with a value of , in good agreement with the experimentally measured value of at high frequency. In the low-frequency region, excess noise is observed due to electronics drifts and interference. For fast detection, the equivalent flux noise is largely dominated by the value in the high frequency limit. For magnetic field detection, the sensitivity can be expressed as , yielding .
The detection sensitivity obtained above is partly limited by the large ratio , as imposed by , required to initialize the qubit by energy relaxation. To reduce the overhead time, we introduce another measurement scheme in which we use the correlator as the detector output, with and two consecutive measurement results at steps and . This is motivated by the fact that an ideal projective readout prepares the qubit in an energy eigenstate. The measurement result is random. However, the product only depends on qubit evolution if energy relaxation is neglected. Our measurement has a limited efficiency and the projection fidelity is lower for due to energy relaxation during measurements lupascu_2007_1 (); picot_2008_rel-meas-qb (). The repetition time of the sequence is experimentally optimized to balance two competing effects: a long results in additional qubit relaxation, which reduces qubit projection when ; a short leads to additional decoherence of the qubit, presumably due to photon number fluctuations in the resonator schuster_2005_1 (). We obtain an optimum detection efficiency for and . We calculate the magnetic field noise referred to detector output using , similar to the scheme based on qubit reset using energy relaxation. The conversion factor is determined from the measurement shown in Fig. 4e. The magnetic field noise is shown in Fig. 4f. A significant improvement of sensitivity is achieved compared to the result obtained using qubit reset by relaxation, as shown in Fig 4c. The equivalent flux noise, averaged over the full frequency interval, is . Excess noise is observed at low frequency as well, however the magnitude is significantly lower than for the data shown in Fig. 4c, due to the fact that low frequency fluctuations in the detection system are removed by using the correlations. The magnetic field detection sensitivity reaches . The improvement in detection efficiency is due primarily to the reduction in duty cycle.
These results establish the PCQ as an ultrasensitive AC magnetic field detector. In the following a discussion is given of how our detector compares with other types of sensors. We focus on the comparison with DC-SQUIDs and atom-based detectors. For DC-SQUIDs, the proper figure of merit related to magnetometry applications is the energy sensitivity where is the loop inductance of the DC-SQUID clarke_2004_1 (). This same figure of merit is adapted to the PCQ, since similar magnetic field coupling methods can be used. In DC-SQUIDs, is dominated at low frequency by flux noise and reaches a constant value in the high frequency region, which is taken as the relevant figure for noise. By energy sensitivity, our detector, with a loop inductance , compares favorably with the DC-SQUIDs in awschalom_2008_LowNoiseSQUID () and wellstood_1989_HotElectronSquids () operated at temperatures of and respectively. We also note that optimization of the energy sensitivity in our case can in principle be done by increasing the loop inductance, as flux noise was observed in general not to scale up with loop size, an aspect favourable for magnetometry clarke_2004_1 ().
For a comparison with atomic magnetometers and NV center based detectors the most adapted figure of merit is the quantity , which combines the sensitivity and the detector volume budker_2007_AtomicMagnReview (). Atomic magnetometers based on vapour cells have very high sensitivity, achieved usually with volumes of the order of , when numbers of the order of dang_2010_MagntoHighSens (); kominis_2003_SubfemtoteslaMagn () are reached. When extrapolated to volumes of , corresponding to the PCQ detector used in this work, the ultimate theoretical limit to sensitivity is of the order of shah_2007_SubPicoTeslaMagn (). More recently, magnetic field detection based on cold atoms has been explored as well wildermuth_2005_BecMagnetometry (); vengalattore_2007_HRMagnBEC (). Using a Bose-Einstein condensate (BEC), a sensitivity of for a measurement area of was obtained in wildermuth_2005_BecMagnetometry (). The fundamental limit for sensing using a BEC with a resolution of a few micrometers wildermuth_2006_BECElectricAndMagnSensing () is in the range. We note that atomic magnetometers operate typically at low frequency, below 1 kHz; methods exist to extend the operation frequency to hundreds of kHz lee_2006_subfemtoRFMagn (). NV centers in diamond taylor_2008_NVDiamondMagn () have recently emerged as an ultrasensitive method for magnetometry. They combine the advantage of the possibility to work at room temperature and a spatial resolution that can be changed from the nanometer range (for single NV center operation) maze_2008_NanoMagnSingleSpin (); balasubramanian_2008_SingleSpinAmbient (); deLange_2011_SingleSpinSensingMultipulse () up to large scale by using a spin ensemble. Decoherence due to paramagnetic impurities limits the flux detection efficiency to , optimal for AC fields at frequencies of the order of 100 kHz taylor_2008_NVDiamondMagn (), which results in a sensitivity of for a detection volume of the order of .
Detection of magnetic fields using the PCQ was demonstrated here for AC magnetic fields at . Detection over a wide range of frequencies is possible. At a given frequency , optimizing the detection sensitivity requires tuning of the magnetic moment of the qubit. This can be achieved in situ by changing the qubit transition frequency . The optimal sensitivity for a PCQ is calculated here with a set of realistic assumptions on parameters and feasible improvements of control and decoherence. Firstly, the persistent current and minimum energy level splitting are taken to have values as the PCQ in this experiment. Secondly, ideal projective measurements are used; projective measurements nearly reaching perfect fidelity have been demonstrated for superconducting qubits (see eg lupascu_2007_1 ()). Thirdly, the duty cycle () is set equal to one. This can be achieved by shortening readout times, as enabled by nearly quantum limited amplifiers castellanosbeltran_2008_ParAmp (); bergeal_2010_PhasePresNearQLimit (), and by replacing spin-echo sequence with more complex control pulse schemes deLange_2011_SingleSpinSensingMultipulse (); bylander_2011_noisePCQ (); taylor_2008_NVDiamondMagn (). Finally, qubit pure dephasing is assumed to be limited by 1/f flux noise (see yoshihara_2006_1 ()), with a spectral density given by , which is the value that would explain the observed spin echo decay time at in our experiment if flux noise was the only noise contribution. This level of flux noise is larger than measured values in smaller area superconducting rings wellstood_1987_dcsquidnoise (); yoshihara_2006_1 (); bylander_2011_noisePCQ (). It is very likely that in our experiment the flux noise is significantly lower than this upper bound and that charge noise plays a major role, due to the low Josephson energy in this device. Straightforward changes in design will allow reducing the influence of charge noise to negligible levels. With these assumptions taken into account, the calculated optimal sensitivity is plotted in Figure 5 for a PCQ with an energy relaxation time of (attained in bylander_2011_noisePCQ ()) and for the case where relaxation is neglected. This calculation shows that with respect to the experimental results reported here, more than one order of magnitude of improvement is possible by feasible changes of the experimental setup. The detection efficiency decreases with frequency, due to the ultimate limit imposed by 1/f noise. Nevertheless, this detector has a very high intrinsic sensitivity for measurements in the range from tens of kHz to tens of MHz. Possible future developments on increase of coherence times of superconducting qubits will increase the efficiency and the useful frequency range even further.
In conclusion, we demonstrated a high sensitivity magnetometer based on an artificial atom. The high magnetic field sensitivity combined with the micron spatial resolution is relevant to applications such as detection of electron spin resonance, scanning probe microscopy, and sensitive current and voltage amplifiers clarke_2004_1 (). This detector is particularly interesting for exploring the dynamics of quantum systems at low temperatures with minimal backaction. The results here illustrate the potential that artificial quantum systems have for quantum sensing.
The PCQ presented in this work is realized using a two-step fabrication process. In the first step, a resist layer is applied on a silicon wafer and patterned using optical lithography, to define all the device elements except the qubit. An Aluminum layer with a thickness of 200 nm is evaporated after resist developing and the step is finalized using lift-off. The second layer, which contains the qubit and the connections to the central line of the coplanar waveguide resonator, is realized using standard shadow evaporation of aluminum. The Josephson junctions are formed by two aluminum layers, with thickness 40 nm and 65 nm respectively, separated by an in-situ grown thin aluminum oxide layer.
The experiments are performed in a dilution refrigerator, at a temperature of 43 mK. The device is placed inside a copper box, connected to a printed circuit board by wire bonding. Connections to transmission lines are done using microwave launchers on the printed circuit board. Magnetic shielding is implemented using three layers of high magnetic permeability material. The sample is connected to room temperature electronics using coaxial cables, which include various filter, isolation, or amplification sections. The signal at the output port of the resonator is amplified using a low-noise high electron mobility transistor (HEMT) amplifier with a noise temperature of 4 K.
Readout and control pulses are implemented using modulation of continuous wave signals produced by synthesizers. Modulation signals are produced using arbitrary waveform generators with a time resolution of 1 ns and 4 ns for control/readout pulses respectively. The signal at the output of the resonator, amplified using the HEMT amplifier, is further amplified using an amplification chain at room temperature, demodulated, and digitized. The average of each readout output pulse is performed and recorded for each repetition. The time series of the digital measurement output are used to extract average quantities and the noise power spectral density.
Magnetic field biasing of the qubit is performed using a centimeter size coil attached to the copper box and fed by a current produced by a high stability current source.
Author contributions: MB and AL designed the experiment; MB fabricated the qubit/resonator device, conducted the experiment, and analyzed the data; JLO and FO contributed to the development of the experimental set-up; CD contributed to the development of software for data acquisition; AL and MB wrote the manuscript; all the authors discussed the data and commented on the manuscript.
Acknowledgements: We are grateful to Mohammad Ansari, Jay Gambetta, Pieter de Groot, Seth Lloyd, Britton Plourde, and Frank Wilhelm for discussions. We acknowledge support from NSERC through Discovery and RTI grants, Canada Foundation for Innovation, Ontario Ministry of Research and Innovation, and Industry Canada. AL is supported by a Sloan Fellowship and JLO is supported by a WIN scholarship.
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